Free energy variation with radius

In a free vortex the energy per unit mass of fluid is constant, and thus a free vortex is inherently stable. The variation of pressure with radius is obtained by differentiating equation 2.44 with respect to radius at constant depth z to give ... 

The value of rcnticat is about 6-10 A (0.6-1 nm) for most materials, which indicates that the droplet of critical size contains between 50 and 100 atoms or molecules. Figure 3.13 shows the variation of the total free energy with the radius of the condensed particle. 

FIGURE 2.6. (a) Phase diagram for a binary system, (b) Variation of the free energy G of a nucleus with its radius r. 

The variation of the Helmholtz energy during dispersion of a homogeneous metal with volume V to particles with radius r can be expressed in Eq. (8-2) where a is the increase in the system s free energy due to the formation of a unit of new interface surface. 

The next question is whether this bubble is stable or not. The following equation shows the variation of the Gibbs free energy with bubble radius for an embryo formed in a superheated liquid due to a density fluctuation ... 

Fig. 3.25 Schematic of variation in the free energy versus radius for a spherical droplet. Some critical size must be exceeded before a nucleus becomes stable. Reproduced with permission from [111]. Copyright 1950, American Chemical Society...

The application of direct variational method for solution of Euler-Lagrange equation is similar to that in Sect. 3.2.2.1. It ailows to obtain the free energy in the form of polarization power series with the coefficients dependent on average particles radius and the parameters of Euler-Lagrange equation (3.45). In particular, surface polarization Pd in the boundary conditions leads to appearance of built-in field Ecyi(R), which can be written as... 

Fig. 7.1. Variation of free energy of formation of a nucleus with radius.

Following the aims of this paper, new sample calculations for the ground-state energy evolution and orbital parameters for O, Si and Ge are presented in Table 4 for selected cavity radii and a barrier height Vo = 1 hartree. The energy shift relative to the free-atom condition is also shown for each case. Comparison with the values obtained for the impenetrable box [Table 2] indicates a less steep variation as the cage radius is reduced, which reflects... 

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